Bifurcation analysis of nonlinear milling process with tool wear and process damping: Sub-harmonic resonance under regenerative chatter

• Milling process including cutting force nonlinearities, tool wear and process damping.
• Non-autonomous delayed parametrically excited equations under sub-harmonic resonance.
• Occurrence of cyclic-fold, symmetry breaking, period doubling and Hopf bifurcations.
• Investigation of jump phenomenon and identification of bi-stability and tri-stability intervals.
• Prediction of periodic, quasi-periodic and chaotic behavior of the limit cycles.

In this paper, bifurcation analysis is performed for the nonlinear milling process under sub-harmonic resonance and regenerative chatter, with tool wear and process damping effects. Multiple-scales approach is used to construct analytical approximate solutions for non-autonomous parametrically excited equations of the system with time delay terms. The new bifurcation parameters are the detuning parameter (deviation of the tooth passing frequency from three times of the chatter frequency), damping ratio (affected by process damping) and tool wear width. Jump phenomenon and multi-values responses are observed in the first order solution under sub-harmonic resonance condition. Periodic, quasi-periodic and chaotic behaviour of the limit cycles are predicted in the presence of regenerative chatter. Change of the detuning parameter leads to the cyclic-fold (tangent) and secondary Hopf (Neimark) bifurcations. It is observed that as damping (affected by process damping) varies, cyclic-fold, secondary Hopf and supercritical symmetry-breaking (pitchfork) bifurcations occur. For the variation of tool wear width, period-doubling (flip), symmetry-breaking and secondary Hopf bifurcations occur. Moreover, for slight values of damping or large values of tool wear width, chaotic behaviour is dominant.